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LaTeX is a program written in the programming language TeX. (In
the same sense, any LaTeX document is also a program, which is
designed to run ‘alongside’, or ‘inside’ LaTeX, whichever metaphor
you prefer.)
Plain TeX is also a program written in the programming language
TeX.
Both exist because writing your documents in ‘raw’ TeX would
involve much reinventing of wheels for every document. They both
serve as convenient aids to make document writing more pleasant:
LaTeX is a far more extensive aid.
LaTeX is close to being a superset of Plain TeX. Many documents
designed for Plain TeX will work with LaTeX with no more than
minor modifications (though some will require substantial work).
Interpretation of any (La)TeX program involves some data-driven
elements, and LaTeX has a wider range of such elements than does
Plain TeX. As a result, the mapping from LaTeX to Plain TeX
is far less clear than that in the other direction — see a later
answer about translating to Plain TeX.
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URL for this question: http://www.tex.ac.uk/cgi-bin/texfaq2html?label=LaTeXandPlain
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This is FAQ version 3.20, last modified on 2010-08-17.